1. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dieter Jaksch (University of Oxford, UK) Bose-Einstein condensates and optical lattices (applications) Dieter Jaksch University of Oxford EU networks: OLAQUI, QIPEST

2. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Ramping an optical lattice S.R. Clark and D.J, Phys. Rev. A 70, 043612 (2004)

3. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. System By changing the laser intensity we go slowly from a shallow lattice U¸J to a deep lattice with UÀJ Time dependent Hamiltonian

4. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Slow dynamics We consider “slowly” ramping the lattice for Eigenvalues of the single particle density matrix

5. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Correlation length cut-off length and momentum distribution width Slow dynamics cont. Define a correlation cut-off length And also consider the momentum distribution width Starting from the MI ground state at t=60ms yields similar results.

6. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. We also computed the correlation cut-off speed for Slow dynamics cont. Note that neither correlation speed exceeds the instantaneous tunnelling speed. (i)= dynamically driven (ii)= MI ground-state Consistent with the growth of correlations being caused by a single atom hopping to the centre. Greiner et al Nature (2002) Batrouni et al PRL (2002) tctc

7. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Replace the latter half with rapid linear ramping of the form Fast dynamics We considered between 0.1 ms and 10 ms. Focussing on where is the total ramping time.

8. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Here we plot (a) the final momentum distribution width for each rapid ramping. Fast dynamics cont. The fitted curve is a double exponential decay with The steady state SF width is acquired in approx. 4 ms. For the 8 ms ramping we plot (b) the correlation speed. Rapid restoration explicable with BHM alone, and occurs in 1D. Higher order correlation functions are important – how do they contribute?.

9. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Excitation spectrum of a 1D lattice S.R. Clark and DJ, New J. Phys. 8, 160 (2006)

10. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. System Periodically changing the laser intensity induces periodic changes in U, J. Time dependent Hamiltonian V 0 = V 0 +  cos(  t)

11. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Probing the excitation spectrum The lattice depth is modulated according to where ideally A is small enough to stay in the regime of linear response. Linear response probes the hopping part of the Hamiltonian; in first order and in second order quadratic response The total energy absorbed by the system is In the experiment A¼0.2 and response calculations thus not applicable

12. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Exact calculation Spectrum for U/J=20 for commensurate filling Vertical lines denote major matrix elements from ground state (i) (ii) (iii)

13. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Exact calculation Spectrum for U/J=4 for commensurate filling Vertical lines denote major matrix elements from ground state How does the energy spectrum change as a function of U/J?

14. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Validity of linear response Small system, comparison to exact calculation Linear response: red curve Exact calculation: blue curve

15. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Numerical simulations Using the TEBD algorithm of G. Vidal to obtain results for larger system of M=40 latttice sites and N=40 atoms for no trap and M=25, N=15 with trap NO trapHarmonic trap

16. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Signatures of SF-MI transition Harmonic trap NO trap Energy in 3U  4U peak Centre position of U peak

17. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Comparison with the experiment We obtain very good agreement with the experimental data broad spectrum in the superfluid region split up into several peaks when going to the MI regime Shift of the MI peaks from U by approximately 10% Differences compared to the experiment Different relative heights of the peaks Transition between SF and MI region at a different value of U/J Signatures in peak height not visible in the experiment exp. Stoferle et al PRL (2004)

18. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Loading and Cooling A. Griessner et al., Phys. Rev. A 72, 032332 (2005). A. Griessner et al., Phys. Rev. Lett. 97, 220403 (2006).

19. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Interband transitions by spontaneous emission of a phonon (particle like) Interband transition by exciting a Fermi gas atom Emission of a particle into background Phonon  FF

20. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Intraband interactions BEC background gas at finite temperature The BEC is deformed by the presence of the lattice atoms Deformation energy E P The deformation affects nearby lattice sites; attractive potential V ij Only elastic collisions are energetically allowed These cause dephasing of the lattice wave function They also lead to thermally induced incoherent hopping Phonon

21. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading and cooling schemes? Can we combine irreversible processes with repulsive and/or Fermi blocking for irreversible loading schemes? Spontaneous emission of photons Large momentum kick  heating Large energies  no selectivity Reabsorption  heating Spontaneous emission of phonons Loading of atoms into lattices Cooling within the lowest Bloch band Destroy spatial correlations Mediate interactions between sites PHOTONS N Phonon A.J. Daley, et al. Phys. Rev. A 69, 022306 (2004).

22. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Current loading methods Adiabatic loading in one sweep with almost no disorder Arrange atoms by repulsion between bosons Arrange atoms by Fermi blocking.. Supefluid Mott insulator D. J., C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39 (2002). L. Viverit, C. Menotti, T. Calarco, A. Smerzi, Phys. Rev. Lett. 93, 110401 (2004).. Fermi gas Single occupancy

23. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Possible Improvements Defect suppressed latticesMeasurement based schemes U aa |ai U bb |bi  irregular  regular filling, i.e., mixed state  pure state  cooling P. Rabl, A. J. Daley, P. O. Fedichev, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 91, 110403 (2003). Selective shift operations to close gaps J. Vala, A.V. Thapliyal, S. Myrgren, U. Vazirani, D.S. Weiss, K.B. Whaley, Phys. Rev. A 71, 032324 (2005). Selective measurements of double occupancies G.K. Brennen, G. Pupillo, A.M. Rey, C.W. Clark, C.J. Williams, Journal of Physics B 38, 1687 (2005).

24. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Initialization of a fermionic register We consider an optical lattice immersed in an ultracold Fermi gas a) Load atoms into the first band b) incoherently emit phonons into the reservoir c) remove remaining first band atoms A. Griessner et al., Phys. Rev. A 72, 032332 (2005).

25. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Coordinate space  |ai n=1 n=0 Linear sweep to load the whole lattice FF Energy Filled Fermi sphere |bi k NkNk We change the laser detuning dynamically a)Sweep through the Fermi sea to load the first band b)Spontaneously emit phonons c)Empty remaining atoms by tuning above the Fermi sea   Not drawn to scale

26. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Loading regimes Fast loading regime: Motion of atoms in background gas is frozen Simple Rabi oscillations Slow loading regime: Background atoms move significantly during loading Coherent loading Dissipative transfer

27. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Result F 0 is the filling in the lower Bloch band F 1 is the filling in the first excited Bloch band

28. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Cooling by superfluid immersion Lattice immersed in a BEC Atoms with higher quasi-momentum q are excited They decay via the emission of a phonon into the BEC They are collected in a dark state in the region q¼0 Analysis using an iterative map in terms of Levy statistics A. Griessner et al., Phys. Rev. Lett. 97, 220403 (2006).

29. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Excitation steps E j In analogy to Raman cooling schemes we choose square pulses with Rabi frequency  j and duraction  j =  /  j. This leads to an excitation probability P(q) shown right. With increasing pulse duraction the region of excitation is narrowed down. All momenta q except those with q¼0 are excited. By using Blackman pulses a more box like (inverted Fermi profile) of excitation probabilities can be obtained.

30. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dissipative steps D Decay according to master equation with Liouvillian (in the momentum basis) This approach is valid for a single particle only as it neglects quantum statistics For many particles: Quantum Boltzmann master equation (QBME) Define occupation vectors Project the density operator onto its diagonal elements We obtain the QBME as the evolution equation for the w m

31. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Results J 0 ¼ 100Hz and this corresponds to T ¼ 5nK  we can reach the few pK regime Single particle, comparison with Levy statistics result Bosons (black) and Fermions (red) cooling Fermion momentum distribution during cooling Single particle momentum distribution during cooling

32. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Cooling atomic patterns

33. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Cooling atoms to the lowest energy sites The existing atoms are compacted, removing the empty sites. These cooling ideas can also be used to redistribute existing atoms, cooling them to the lowest energy sites: A.J. Daley et al., in progress

34. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Cooling mechanism J J Sites offset by  Off-resonant Raman process couples lowest Bloch band to upper band Tunnelling in upper band J (tunnelling in lowest band small) Cooling from excited motional states via phonon emission We must choose the  and  appropriately to avoid populating the excited motional states.     

35. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Cooling Mechanism  J J Sites offset by  Off-resonant Raman process couples lowest Bloch band to upper band Tunnelling in upper band J (tunnelling in lowest band small) Cooling from excited motional states via phonon emission We must choose the  and  appropriately to avoid populating the excited motional states.  e.g., , J ¿ ,   ¿ Gamma:  Sideband Cooling!!!  Cooling /  (  J/  ) 2 / [(  -  ) 2 +  2 /4)] Heating /  (  J/  ) 2 / [(  +  ) 2 +  2 /4)]

36. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Timescale Limitations: 1/  ¿ J BUT J is tunelling in upper band Cooling Mechanism J J      Further Analysis: Quantum Boltzmann Master Eq. Quantum Trajectories Simulation [see D. Jaksch PRA 56, 575 (1997)]

37. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Irreversible loading of optical lattices Dephasing in the lowest band M. Bruderer and D. Jaksch, New J. Phys. 8, 87 (2006).

38. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Realization of impurities in BECs Two species mixtures of distinct atoms A deep optical lattice realizes atomic quantum dots Other species is a BEC in 1D, 2D or 3D Impurities are assumed to interact independently (dÀ  The impurity couples to a fluctuating particle density The condensate density operator The impurity is trapped inside the BEC M. Bruderer and D. Jaksch, New J. Phys. 8, 87 (2006). BEC width  (A. Recati et al. PRL 2005)

39. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dephasing of a quantum dot Density-density interactions between impurity and BEC The resulting dephasing is determined by the fluctuations of the superfluid phase We calculated effects of density fluctuations at temperature T. For t! 1 A SAT in a 1D BEC cannot be used as a qubit

40. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. BEC Quantum dot as a quantum probe (I) The SAT can be used as quantum probe (of variable size) to detect properties like temperature of a BEC, Tonks gas, Mott insulators via Ramsey interference. 3D 2D 1D

41. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Probing the dimensionality of the BEC Dephasing as a function of dimensionality of the BEC  /c  /c  /c

42. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Probing the temperature of a BEC Dephasing as a function of the temperature of a BEC 1D2D  /c  /c  /c  /c  /c  /c

43. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Summary Optical lattices Can cleanly realize interesting quantum dynamics Large degree of experimental control over experimental parameters Geometry Temperature Energies (kinetic, potential and interaction) Decoherence (by immersion) Study the role of quantum correlation Nucleation of a superfluid Effects of Non-adiabaticity ramping a lattice Excitation spectra in the non-linear regime Loading and cooling via phonon emission Dephasing and polaron formation in the presence of background gas …